11 THE SPECTRAL THEORY OF HUBERT 399 (15.11 .7) For N to be self-adjoint (resp. unitary) it is necessary and sufficient that Sp(N) c R (resp. Sp(JV) c:U).IfHis self-adjoint, then inf(SpCff)) = inf (H x \ x), (15.11.7.1) II*""1 sup(Sp(#)) = sup (H-x|x), ll*ll*i (15.11.7.2) ||H||= sup|(H-x|x)|. ll*ll-i In the first assertion, we have already seen (15.4.12) that the conditions are necessary. To show that they are sufficient, it is enough (by virtue of (1 5.1 1 .5)) to prove them for the Nn\ and this is immediately done, because, when Nn is identified with multiplication by the class of lc in L^(Sp(Nt)9 /4n), the operator N* is identified with multiplication by the class of the function Ł*-->Ł To prove (15.11.7.1), it is enough to show that, for a self-adjoint operator H to be such that (H x \ x) ^ 0 for all x e E (in which case we say that H is positive and we write H ^ 0), it is necessary and sufficient that Sp(/f) c R+ . In view of (15.11.5) and the relation (15.11.7.3) (H-xIx^KH.-xJxJ 71 (with notation analogous to that of (15.11.3)) we are reduced to proving the assertion for simple self-adjoint operators H* . If we identify Hn with multipli- cation by the class of lc in Lc(Sp(Hn), /*), what we have to prove is that Sp(Hn) <= R+ if and only if JV(0 4UŁ) ^ 0 for every function /Ł 0 in ^rc(Sp(/fn)). Now, if M is the intersection of Sp(Hn) with the complement ] oo, 0[ of R+ in R, then the relation M ^0 would imply /JB(M) > 0 (15.1.14). Since ] oo.0[ is the union of the intervals ] oo, 1/Ť], there would exist m > 0 such that and consequently f C^CO 4*n(C) ^ -a/w < 0, contrary to hypothesis. Finally, the relation (15.11.7.2) follows from (15.11.7.1) and (15.4.14,1), because the spectral radius of H is equal to the larger of |inf (Sp(^T))|, |sup(Sp(ff))|. (15.11.8) (i) For each function fe ^rc(Sp(N)), the spectrum of f(N) is contained in f(Sp(N)) (closure in C), and (15.11.8.1) ||/(N)|| g sup |/(C)|.ntegers n such that a